How do I unwrap a cylindrical object?
Looking at your image, it looks like your seams aren't marked correctly. You should have an unbroken seem (red line in edit mode) running up the whole of the side of the mesh. This line should connect to a ring seam (a full red ring around the mesh) on each end, that separates the ends from the sides.

As lemon mentioned in the comments, you may also need to follow the instructions in How to make a square grid for UV maps to get even quads in the unwrap.

It may be worth looking at some basic uv unwrapping tutorials.